Height and Distance — NDA Mathematics PYQs

A plane is observed to be approaching the airport. It is at a distance of $10\text{ km}$ from the point of observation and makes an angle of elevation of $67\cdot 5^{\circ }$. What is the height of the plane above the ground ?

Consider the following for the two (02) items that follow :
The top ($M$) of a tower is observed from three points $P$, $Q$ and $R$ lying in a horizontal straight line which passes directly along the foot ($N$) of the tower. The angles of elevations of $M$ from $P$, $Q$ and $R$ are $30°$, $45°$ and $60°$ respectively. Let $PQ=a$ and $QR=b$.

What is $PN$ equal to?

Consider the following for the two (02) items that follow :
The top ($M$) of a tower is observed from three points $P$, $Q$ and $R$ lying in a horizontal straight line which passes directly along the foot ($N$) of the tower. The angles of elevations of $M$ from $P$, $Q$ and $R$ are $30°$, $45°$ and $60°$ respectively. Let $PQ=a$ and $QR=b$.

What is $MN$ equal to?

A man at $M$, standing $100$ m away from the base ($P$) of a chimney of height $50$ m, observes the angle of elevation of the highest point ($Q$) of the smoke to be $45°$. The highest point of the chimney is at $R$. Further $P$, $R$ and $Q$ are in a straight line and the straight line is perpendicular to $PM$. What is the angle $RMQ$ equal to?

*Directions : Consider the following for the next two (02) items that follow :

A flagstaff $20\text{ m}$ long standing on a pillar $10\text{ m}$ high subtends an angle ${\tan }^{-1}(0\cdot 5)$ at a point $P$ on the ground. Let $\theta$ be the angle subtended by the pillar at this point $P$.*

If $x$ is the distance of $P$ from bottom of the pillar, then consider the following statements :

1. $x$ can take two values which are in the ratio $1:3$
2. $x$ can be equal to height of the flagstaff

Which of the statements given above is/are correct ?

*Directions : Consider the following for the next two (02) items that follow :

A flagstaff $20\text{ m}$ long standing on a pillar $10\text{ m}$ high subtends an angle ${\tan }^{-1}(0\cdot 5)$ at a point $P$ on the ground. Let $\theta$ be the angle subtended by the pillar at this point $P$.*

What is a possible value of $\tan \theta$ ?

A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height $h$. At a point on the plane the angles of elevation of the bottom and top of the flagstaff are $\theta$ and $2\theta$ respectively. What is the height of the tower?

The shadow of a tower becomes $x$ metre longer, when the angle of elevation of sun changes from $60^{\circ }$ to $\theta$. If the height of the tower is $\sqrt{3}x$ metre, then which one of the following is correct?

Directions : Consider the following for the next three (03) items that follow:
$ABC$ is a triangular plot with $AB=16\text{ m}$, $BC=10\text{ m}$ and $CA=10\text{ m}$. A lamp post is situated at the middle point of the side $AB$. The lamp post subtends an angle $45^{\circ }$ at the vertex $B$.

What is the height of the lamp post?

Directions : Consider the following for the next three (03) items that follow:
There are two points $P$ and $Q$ due south of a leaning tower, which leans towards north. $P$ is at a distance $x$ and $Q$ is at a distance $y$ from the foot of the tower ($x>y$). The angles of elevation of the top of the tower from $P$ and $Q$ are $15^{\circ }$ and $75^{\circ }$ respectively.

At what height is the top of the tower above the ground level?